Accurate geological modelling of features such as faults, fractures or erosion requires grids that are flexible with respect to geometry. Such grids generally contain polyhedral cells and complex grid-cell connectivities. The grid representation for polyhedral grids in turn affects the efficient implementation of numerical methods for subsurface flow simulations. It is well known that conventional two-point flux-approximation methods are only consistent for K-orthogonal grids and will, therefore, not converge in the general case. In recent years, there has been significant research into consistent and convergent methods, including mixed, multipoint and mimetic discretisation methods. Likewise, the so-called multiscale methods based upon hierarchically coarsened grids have received a lot of attention. The paper does not propose novel mathematical methods but instead presents an open-source Matlab® toolkit that can be used as an efficient test platform for (new) discretisation and solution methods in reservoir simulation. The aim of the toolkit is to support reproducible research and simplify the development, verification and validation and testing and comparison of new discretisation and solution methods on general unstructured grids, including in particular corner point and 2.5D PEBI grids. The toolkit consists of a set of data structures and routines for creating, manipulating and visualising petrophysical data, fluid models and (unstructured) grids, including support for industry standard input formats, as well as routines for computing single and multiphase (incompressible) flow. We review key features of the toolkit and discuss a generic mimetic formulation that includes many known discretisation methods, including both the standard two-point method as well as consistent and convergent multipoint and mimetic methods. Apart from the core routines and data structures, the toolkit contains addon modules that implement more advanced solvers and functionality. Herein, we show examples of multiscale methods and adjoint methods for use in optimisation of rates and placement of wells.

We analyse and further develop a hierarchical multiscale method for the numerical simulation of two-phase flow in highly heterogeneous porous media. The method is based upon a mixed finite-element formulation, where fine-scale features are incorporated into a set of coarse-grid basis functions for the flow velocities. By using the multiscale basis functions, we can retain the efficiency of an upscaling method by solving the pressure equation on a (moderate-sized) coarse grid, while at the same time produce a detailed and conservative velocity field on the underlying fine grid. Earlier work has shown that the multiscale method performs excellently on highly heterogeneous cases using uniform coarse grids. In this paper, we extend the methodology to nonuniform and unstructured coarse grids and discuss various formulations for generating the coarse-grid basis functions. Moreover, we focus on the impact of large-scale features such as barriers or high-permeable channels and discuss potentially problematic flow cases. To improve the accuracy of the multiscale solution, we introduce adaptive strategies for the coarse grids, based on either local hierarchical refinement or on adapting the coarse grid more directly to large-scale permeability structures of arbitrary shape. The resulting method is very flexible with respect to the size and the geometry of coarse-grid cells, meaning that grid refinement/adaptation can be performed in a straightforward manner. The suggested strategies are illustrated in several numerical experiments.

The bookh as grown out of ac oncerted research effort overt he last decade. We have enjoyedc ollaboration withm anyg ood friends and colleagues on these problems, in particular, Raimund Bürger, GiuseppeC oclite, Helge Dahle,M agne Espedal, Steinar Evje, Harald Hanche-Olsen, RunarHoldahl,T rygveKarper,V egard Kippe, Siddhartha Mishra, Xavier Raynaud, and John Towers, and we use this opportunityt ot hank them for the joyo fc ollaboration. Ourr esearchh as been supported in part by the ResearchC ouncil of Norway. K.-A.L ie gratefully acknowledgess upport fromS imula ResearchL aboratory for parto fh is work on this project.K .H .K arlsen has alsob een supported by an Outstanding Young Investigators Award from the ResearchC ouncilo fN orway. Part of the bookw as writtena sp art of thei nternational researchp rogram on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the NorwegianA cademyo fS cience and Letters in Oslo during the academic year 2008-09. We have established aw eb site, www.math.ntnu.no/operatorsplitting, where we willp ost computer codes in Matlab 1 for thee xamples as well as al ist of errata. Please let us know if youfi nd errors.

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